Feature engineering in neural networks optimization

ABSTRACT

A transitive closure data structure is constructed for a pair of features represented in a vector space corresponding to an input dataset. The data structure includes a set of entries corresponding to a set of all possible paths between a first feature in the pair and a second feature in the pair in a graph of the vector space. The data structure is reduced by removing a subset of the set of entries such that only a single entry corresponding to a single path remains in the transitive closure data structure. A feature cross is formed from a cluster of features remaining in a reduced ontology graph resulting from the reducing the transitive closure data structure. A layer is configured in a neural network to represent the feature cross, which causes the neural network to produce a prediction that is within a defined accuracy relative to the dataset.

TECHNICAL FIELD

The present invention relates generally to a method, system, andcomputer program product for configuring a neural network to producepredictions based on large datasets. More particularly, the presentinvention relates to a method, system, and computer program product forfeature engineering in neural networks optimization.

BACKGROUND

A dataset as used herein is a collection, body, or corpus of data inwhich a data item corresponds to one or more features from a featureset. A feature is an entity that is represented in the data. Forexample, in a dataset that is a snippet from a book in Englishliterature, distinct words can be regarded as features. As anotherexample, in a dataset that includes traffic flow data, days of a weekand hours of a day can be regarded as features.

A feature is not only an entity that is directly or manifestlyrepresented in the data but also an entity that can be inferred from thedata. For example, suppose that a dataset contains product ownershipdata and one data item in the dataset shows that company X owns productY. In the dataset, X and Y are directly represented features (as are thegeneric classes “company” and “products”), but the fact that X also ownsversion 10 of product Y can also be inferred from X owning Y and Yhaving a version 10. Thus, “Y version 10” is also a feature that can beinferred from the dataset.

A feature cross (FC) is a combination or concatenation of two or morefeatures. For example, using the dataset with traffic data as an exampleagain, while ‘Monday’ and ‘2 PM’ might be features in the dataset,‘Monday+2 PM’ (i.e., Monday at 2 PM) is a feature cross formed fromcombining features ‘Monday’ and ‘2 PM. That is, a data item maycorrespond to ‘Monday’ but may or may not correspond to ‘2 PM’; anotherdata item may correspond to ‘2 PM’ but may correspond to ‘Thursday’instead of ‘Monday’. Only a data item that corresponds to both ‘Monday’and ‘2 PM’ would correspond to feature cross ‘Monday+2 PM’.

Given a dataset as input, a neural network (NN) can be configured tomake predictions based on the dataset. For example, given a dataset thatis a snippet from a book in English literature, a neural network can beconfigured to predict whether some specific words will occur together orat some specified distance from one another. As another example, in adataset that includes traffic flow data, a neural network can beconfigured to predict a volume of traffic at a certain hour on a certainday in the future.

An ontology as referred to herein is a graph representation of a datasetin which the features form the vertices and relationships betweenfeatures—whether expressly present in the dataset or inferred therefrom—form the edges. An ontology according to the illustrativeembodiments can have vertices representing features, feature crosses, ora combination thereof.

An Artificial Neural Network (ANN)—also referred to simply as a neuralnetwork—is a computing system made up of a number of simple, highlyinterconnected processing elements (nodes), which process information bytheir dynamic state response to external inputs. ANNs are processingdevices (algorithms and/or hardware) that are loosely modeled after theneuronal structure of the mammalian cerebral cortex but on much smallerscales. A large ANN might have hundreds or thousands of processor units,whereas a mammalian brain has billions of neurons with a correspondingincrease in magnitude of their overall interaction and emergentbehavior.

A neural network can be configured to represent a set of features,feature crosses, or some combination thereof. The illustrativeembodiments recognize that configuring a neural network such that thepredictions are consistent with the dataset and an expected outcome (oran actual outcome if available) is a difficult problem for a variety ofreasons. For example, a size of a neural network is related to a numberof features or feature crosses on which the neural network is expectedto operate.

Again, consider the traffic dataset example. There are only 7 days of aweek features and 24 hours of a day features, a total of 31 features.However, not all hours of days are equally significant for prediction.E.g., one may be more interested in a traffic prediction at 8 PM onFridays rather than at 8 PM on Mondays. However, a total of 7*24=168feature cross is now possible with just two types of features beingcrossed. The illustrative embodiments recognize that a neural networkthat represents 31 features is substantially less complex than a neuralnetwork that is configured for 168 feature crosses.

Furthermore, the illustrative embodiments recognize that a dataset canbe represented as a matrix in which the features form the columns andeach data item is a row. A row gets an entry in a cell under a column ifthe data item of that row relates to the feature of that column. As canbe seen, even the matrix with 31 columns for 7 days and 24 hours can besparsely populated in that not every hour of every day has traffic datathat might be available/useful/meaningful. A matrix of feature crossesbecomes even more sparse as compared to the matrix of the features towhich the feature crosses correspond. E.g., with just day+hour type offeature crosses, the revised matrix now has 168 columns instead of 31,with a row now corresponding to an even smaller percentage of cells inthe matrix. A neural network configured on these 168 example featurecrosses would have to hold a very sparse matrix in memory to perform thecomputations.

In practice, a matrix is often many thousands of columns by manythousands of rows, and a neural network has several layers with eachlayer including thousands of nodes. Training a neural network on adataset causes the weights of these nodes to be computed and adjustedfor bringing the output of the neural network within a desired thresholdof accuracy. The adjustment of weights of a neural network is acomputationally expensive process. Keeping large but sparse matricesrequires undesirably large amounts of memory for the training andoperation of the neural network. The illustrative embodiments recognizethat while feature crossing is desirable to increase the relevance ofthe output of the neural network, feature crossing also causes a matrixto become sparser and increases the complexity and resource requirementsof the neural network.

SUMMARY

The illustrative embodiments provide a method, system, and computerprogram product. An embodiment includes a method that constructs, usinga processor and a memory, a transitive closure data structure for a pairof features represented in a vector space corresponding to an inputdataset, wherein the transitive closure data structure comprises a setof entries corresponding to a set of paths in a graph of the vectorspace, the set of paths comprising all possible paths in the graphbetween a first feature in the pair and a second feature in the pair.The embodiment reduces the transitive closure data structure by removingfrom the transitive closure data structure a subset of the set ofentries such that only a single entry corresponding to a single pathremains in the transitive closure data structure. The embodiment forms afeature cross from a cluster of features remaining in a reduced ontologygraph resulting from the reducing the transitive closure data structure.The embodiment configures a layer in a neural network to represent thefeature cross, wherein the configured layer in the neural network causesthe neural network to produce a prediction that is within a definedaccuracy relative to the dataset.

Another embodiment further computes, using the processor and the memory,for the layer, an upper bound of layer embedding corresponding to thefeature cross. The embodiment computes, for the layer, a lower bound ofthe layer embedding corresponding to the feature cross, wherein a layersize of the layer for layer embedding is based on the upper bound andthe lower bound.

In another embodiment, the layer size is a computed average of the upperbound and the lower bound.

Another embodiment further transforms the dataset into a sparse matrix.The embodiment constructs an ontology graph corresponding to the sparsematrix, wherein the ontology graph comprises a set of verticescorresponding to a set of features in the dataset. The embodimentperforms a forward materialization on a graph, wherein the forwardmaterialization adds an inference vertex to the set of vertices, andwherein the set of vertices including the inference vertex is used inconstructing the transitive closure data structure.

In another embodiment, the single remaining path satisfies an efficiencyrequirement.

In another embodiment, the efficiency requirement comprises a shortestdistance between the first feature and the second feature.

In another embodiment, the efficiency requirement comprises a highestusage between the first feature and the second feature.

In another embodiment, the reducing comprises a transitive reductionalgorithm.

In another embodiment, the reducing the transitive closure datastructure causes a subset of vertices to be removed from the ontologygraph, forming the reduced ontology graph.

Another embodiment further identifies a plurality of clusters in thereduced ontology graph, the plurality of clusters including the cluster,and wherein the cluster comprises at least two features such that acombination of the two features is usable to distinguish between atleast a first portion of the dataset and a second portion of thedataset.

An embodiment includes a computer usable program product. The computerusable program product includes a computer-readable storage device, andprogram instructions stored on the storage device.

An embodiment includes a computer system. The computer system includes aprocessor, a computer-readable memory, and a computer-readable storagedevice, and program instructions stored on the storage device forexecution by the processor via the memory.

BRIEF DESCRIPTION OF THE DRAWINGS

Certain novel features believed characteristic of the invention are setforth in the appended claims. The invention itself, however, as well asa preferred mode of use, further objectives and advantages thereof, willbest be understood by reference to the following detailed description ofthe illustrative embodiments when read in conjunction with theaccompanying drawings, wherein:

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented;

FIG. 2 depicts a block diagram of a data processing system in whichillustrative embodiments may be implemented;

FIG. 3 depicts a block diagram of an example problem that can be solvedwith an illustrative embodiment;

FIG. 4 depicts a block diagram of an example embedding in accordancewith an illustrative embodiment;

FIG. 5 depicts a block diagram of an example application for featureengineering in neural networks optimization in accordance with anillustrative embodiment;

FIG. 6 depicts a materialization of inference in accordance with anillustrative embodiment;

FIG. 7 depicts an output graph resulting from a graph according to anembodiment;

FIG. 8A depicts an placement of an ontology into a vector space;

FIG. 8B depicts the materialization of inferences for candidate featurecrosses; and

FIG. 9 depicts a flowchart of an example process for feature engineeringin neural networks optimization in accordance with an illustrativeembodiment.

DETAILED DESCRIPTION

Neural network training and optimization is a well-recognizedtechnological field of endeavor. The present state of the technology inthis field of endeavor has certain drawbacks and limitations. Theoperations and/or configurations of the illustrative embodiments impartadditional or new capabilities to improve the existing technology inthis technological field of endeavor, especially in the area of featuresselection, feature crossing, and embedding.

A sparse matrix is a matrix in which less than a threshold percentage ofcells are non-null, non-void, non-zero, or some other representation ofan indicator that the cell is occupied or is to be used. A dense matrixis a matrix in which greater than the threshold percentage of cells arenon-null, non-void, non-zero, or some other representation of anindicator that the cell is occupied or is to be used. The illustrativeembodiments recognize that having a dense matrix such that the densematrix is still representative of the data in a sparse dataset, within adefined tolerance of accuracy, is desirable. One way of achieving adense matrix for a dataset is by selectively crossing certain features,omitting certain features or feature crosses, or both.

The illustrative embodiments recognize that selecting appropriatefeatures to cross to form feature crosses that result in a dense matrixis a difficult problem to solve with practical matrices. Presentlyavailable techniques rely on the personal knowledge and expertise ofhumans who specialize in a subject-matter area to select the features tocross from the datasets pertaining to that subject-matter. Judiciousselection of features to cross can result in a dense matrix from asparse matrix. For example, while an all-inclusive feature crossingeffort results in 168 feature crosses in the example traffic dataset,not all 168 feature crosses are really useful for practical purposes. Ahuman skilled in traffic data analysis can selectively determine thatFriday at 8 PM tends to be interesting and should form an feature cross,whereas Mondays at 2 PM and Wednesdays at 3 PM appear to be almostsimilar in the traffic data, so Monday at 2 PM and Wednesday at 3 PMshould be crossed together as one feature cross. The human might decidethat data for Tuesday at 1 AM is negligible or non-existent andtherefore Tuesday at 1 AM should not even be represented in the matrixof feature crosses.

Conversely, crossing too many features or incorrectly related featurescan lead to data loss, inaccurate prediction, or both. For example, Ifone were to cross features 12 AM-5 AM for all weekdays Monday-Friday, 6AM-11 AM for all weekdays Monday-Friday, 12 PM-5 PM for all weekdaysMonday-Friday, and 6 PM-11 PM for all weekdays Monday-Friday, a verysmall and dense matrix with only four feature crosses would indeedresult. However, such a matrix will not be able to differentiate betweenthe characteristics of Monday at 7 PM versus Friday at 7 PM, which couldbe important for analysis and prediction.

Thus, presently, a manual exercise is required with specific skills andknowledge in order to transform a sparse matrix into a useful densematrix. The illustrative embodiments recognize that an automated methodof selecting features to cross, to construct a dense matrix thatcorresponding to a sparse dataset without unacceptable loss of datagranularity or unacceptable loss of accuracy, would be useful. Theillustrative embodiments further recognize that such an automated methodto iteratively improve the dense matrix with a machine learning methodvia a feedback from a production implementation of a neural networktrained on the dense matrix would also be useful.

According to the illustrative embodiments, embedding is a process ofconfiguring a neural network layer with nodes corresponding to thefeature crosses in the dense matrix such that an output of the neuralnetwork is within a defined tolerance of accuracy relative to an outputof another neural network whose layer is configured according to thefeature or feature crosses of the sparse matrix from which the densematrix is formed. A layer size is a number of nodes in a layer where thenodes in that numerosity are trained corresponding to the featurecrosses from the dense matrix. In a general embodiment, the number ofnodes in a layer have n:m correspondence with the number of featurecrosses in the dense matrix (i.e., n nodes per m feature crosses). Inone embodiment, the number of nodes in a layer have 1:1 correspondencewith the number of feature crosses in the dense matrix (i.e., one nodeper feature cross). In another embodiment, the number of nodes in alayer have 1:n correspondence with the number of feature crosses in thedense matrix (i.e., one node per n feature crosses). In anotherembodiment, the number of nodes in a layer have n:1 correspondence withthe number of feature crosses in the dense matrix (i.e., n nodes perfeature cross).

The illustrative embodiments recognize that the larger the layer sizethe higher is the amount of computing and data storage resources forboth training as well as operating, as compared to a smaller layer size.At the same time, the larger the layer size, the more accurate is theoutput of the layer as compared to the output of a layer of a smallersize.

The illustrative embodiments recognize that the feature selections forcrossing and the layer size are also interdependent because layer sizeis dependent on a set of hyperparameters, which include but are notlimited to the feature crosses. For example, if a given layerconfiguration does not produce a desired output or accuracy, ahyperparameter of the neural network has to be adjusted. A feature is ahyperparameter and if an feature cross is changed, the change can causeone or more hyperparameters of the neural network to change. A change ina hyperparameter requires that the neural network be retrained with thenew hyperparameters. Training a neural network with thousands of nodesin several layers with matrices that are thousands of columns wide anddeep is computationally prohibitively expensive. Therefore, the less thehyperparameters of the neural network are disturbed, the lower the costof training and deploying the neural network. Thus, again, automatedsmart selection of features for crossing will be useful.

Furthermore, given a particular dense matrix with a certain densityachieved via forming certain feature crosses, a particular embeddingproduces an output of a certain accuracy. Different embeddings arepossible from the same dense matrix by selecting different ways ofrepresenting the set or subset of feature crosses from the matrix intothe embedding, each producing a different output of a differentaccuracy. Thus, the illustrative embodiments recognize that there existsa range of embeddings for each subset of feature crosses in which theembeddings produce outputs of acceptable accuracy.

The illustrative embodiments recognize that presently no systematic andobjective method exists for selecting an embedding. Generally, humansskilled in the art of neural network training experiment with differentembeddings, at great computational cost, to arrive at an embedding thatis acceptable. The illustrative embodiments recognize that a systematicand objective computation of an upper bound and a lower bound ofembedding for a given dense matrix will be useful.

The illustrative embodiments recognize that the presently availabletools or solutions do not address these needs/problems or provideadequate solutions for these needs/problems. The illustrativeembodiments used to describe the invention generally address and solvethe above-described problems and other related problems by featureengineering in neural networks optimization.

An embodiment can be implemented as a combination of certain hardwarecomponents and a software application. An implementation of anembodiment, or one or more components thereof, can be configured as amodification of an existing neural network training and predictionconfiguration, with a companion software application executing in somecombination of (i) the existing neural network training and predictionconfiguration itself, (ii) a data processing system communicating withthe existing neural network training and prediction configuration overshort-range radio or a local area network (LAN), and (iii) a dataprocessing system communicating with the existing neural networktraining and prediction configuration over a wide area network (WAN).

An embodiment forms an ontology from a given dataset. In other words,the embodiment labels vertices with labels (features) and edges withrelationships found in the dataset. The embodiment further builds theontology by populating additional vertices and edges corresponding toderived or inferred labels and relationships. This operation of buildingthe ontology through inferences is called forward materialization.

An embodiment forms a vector space corresponding to the graph of theontology. A vector space comprises numerical representation of data. Thevector space formed by the embodiment includes the numericalrepresentations of the features that are populated in the graph. Avector space produced by an embodiment is usable for determiningsimilarities between concepts present in the vector space. For example,an embodiment applies a similarity function to determine similaritiesbetween the concepts underlying the features. A concept is anabstraction or generalized representation of a feature. For example,Monday is a feature and Tuesday is a feature, and they are similarbecause they are both manifestations of the concept of days of a weekeven though the dataset may itself not indicate anywhere that Monday andTuesday are days of a week. Concepts are not limited to this trivialexample, and seemingly unrelated features can become linked throughesoteric concepts in an embodiment's similarity analysis. Cosinesimilarity is one non-limiting example of a similarity function that canbe used in an embodiment for the purpose of similarity determination inthe vector space.

An embodiment constructs a transitive closure data structurecorresponding to the vector space. a transitive closure data structureis a data structure that contains information about all the possiblepaths from reaching point B from point A in the vector space. Thetransitive closure data structure includes one or more paths betweenfeatures, inferences, and concepts. An embodiment analyzes a transitiveclosure data structure to determine and select the most desirable pathfrom a plurality of paths between the two entities represented in thetransitive closure data structure. In one embodiment, a path isdesirable if the path is the shortest or efficient of all paths in thetransitive closure data structure. In other embodiments, a path can alsobe desirable if the path is the most used path in the transitive closuredata structure. Generally, any selection criterion can be used to selectone path from a plurality of paths in a transitive closure datastructure.

The embodiment removes the unselected paths from a transitive closuredata structure. Removal of an unselected path from a transitive closuredata structure also removes one or more vertices, edges, or acombination thereof, from the ontology graph. Removal of a vertex oredge from the graph is indicative of a less than a desirable level ofcontribution of the vertex or edge in the graph. In other words, aremoved vertex or edge is deemed to not be contributing to the graph ina meaningful enough way that its removal is unlikely to affect anundesirable (more than a threshold) decrease in an accuracy ofprediction with the dataset after the removal.

Thus, an embodiment performs a graph reduction to form a reduced graphsuch that the reduced graph is still sufficiently representative of theoriginal dataset, but includes features, inferences, and conceptsinterconnected via the most desirable paths in the graph. This manner ofgraph reduction is called a transitive reduction. The transitivereduction of an embodiment can be controlled by the extent to whichinferences are drawn, parameters of the similarity function to findsimilarities and concepts, criteria for selecting a desirable path,removal or non-removal of certain undesirable vertices or edges, or somecombination thereof.

A remaining vertex in the reduced graph is called a discriminative nodebecause it has the ability to discriminate between data items in thematrix and is not redundant, perfunctory, or superfluous. An embodimentidentifies one or more clusters in the remaining vertices in the reducedgraph. A cluster identifies features that can be crossed to form usefulfeature crosses that will not cause an unacceptable level of data loss,loss of accuracy, or both. Each identified feature cross has an upperand lower bounds of cells in the dense matrix. The upper bound is acomputation of all positive instances records for each feature thatcontributes to the cluster irrespective of the transitive reduction costformula. For example, if in the traffic data example, we did not haveany data for Monday at 2 AM, those vertices and edges would fall offduring the reduction without any loss of information. Therefore, at thehighest bound, the graph is the original graph minus the features thatdo not contribute to the graph (have no paths through them). In thetraffic data example, this might still reduce the graph from the 168possible feature crosses to, e.g., 142 or some other smaller number. Ina more selective embodiment, Monday at 2 AM might have less than aminimal number of data items, or less than a statistical distributioncutoff number of data items and still might get dropped off, in whichcase, the graph is even smaller but still without significant loss ofdata or accuracy. Outlier data items and corresponding features can beexcluded in this manner.

The lower bound is a function output of all discriminative nodes andtheir distinct values. Here, because all undesirable paths wereeliminated, the elimination may have removed some feature crosses thatwere meaningful, just not the most meaningful. In other words, thereduction caused the graph to include only the highly discriminatingnodes in the vector space, which invariably would cause lessdiscriminating features, feature crosses, and the corresponding dataitems to be eliminated. In the example traffic dataset, the lower boundcould therefore be much lower, e.g., 56 or some smaller number, but withthe associated loss of data and/or accuracy.

Thus, an embodiment produces the high and low bounds for the embeddings.An embodiment performs a dimensionality reduction in embedding using theupper and lower bounds of an identified feature cross. For example, oneembodiment takes a statistical average of the high and low bounds andcomes up with a number that forms the layer size for an embedding.

The manner of feature engineering in neural networks optimizationdescribed herein is unavailable in the presently available methods inthe technological field of endeavor pertaining to neural networkconfiguration and operation for predictions. A method of an embodimentdescribed herein, when implemented to execute on a device or dataprocessing system, comprises substantial advancement of thefunctionality of that device or data processing system in systematicallyand objectively optimizing the dataset to be used for training a neuralnetwork, and producing a neural network in which the layer size isoptimized to produce predictions of desirable accuracy withsignificantly reduced computing resources.

The illustrative embodiments are described with respect to certain typesof data items, datasets, features, graphs, feature crosses, algorithms,formulae, functions, reductions, neural networks, layers, nodes,relationships, clusters, data structures, devices, data processingsystems, environments, components, and applications only as examples.Any specific manifestations of these and other similar artifacts are notintended to be limiting to the invention. Any suitable manifestation ofthese and other similar artifacts can be selected within the scope ofthe illustrative embodiments.

Furthermore, the illustrative embodiments may be implemented withrespect to any type of data, data source, or access to a data sourceover a data network. Any type of data storage device may provide thedata to an embodiment of the invention, either locally at a dataprocessing system or over a data network, within the scope of theinvention. Where an embodiment is described using a mobile device, anytype of data storage device suitable for use with the mobile device mayprovide the data to such embodiment, either locally at the mobile deviceor over a data network, within the scope of the illustrativeembodiments.

The illustrative embodiments are described using specific code, designs,architectures, protocols, layouts, schematics, and tools only asexamples and are not limiting to the illustrative embodiments.Furthermore, the illustrative embodiments are described in someinstances using particular software, tools, and data processingenvironments only as an example for the clarity of the description. Theillustrative embodiments may be used in conjunction with othercomparable or similarly purposed structures, systems, applications, orarchitectures. For example, other comparable mobile devices, structures,systems, applications, or architectures therefor, may be used inconjunction with such embodiment of the invention within the scope ofthe invention. An illustrative embodiment may be implemented inhardware, software, or a combination thereof.

The examples in this disclosure are used only for the clarity of thedescription and are not limiting to the illustrative embodiments.Additional data, operations, actions, tasks, activities, andmanipulations will be conceivable from this disclosure and the same arecontemplated within the scope of the illustrative embodiments.

Any advantages listed herein are only examples and are not intended tobe limiting to the illustrative embodiments. Additional or differentadvantages may be realized by specific illustrative embodiments.Furthermore, a particular illustrative embodiment may have some, all, ornone of the advantages listed above.

With reference to the figures and in particular with reference to FIGS.1 and 2, these figures are example diagrams of data processingenvironments in which illustrative embodiments may be implemented. FIGS.1 and 2 are only examples and are not intended to assert or imply anylimitation with regard to the environments in which differentembodiments may be implemented. A particular implementation may makemany modifications to the depicted environments based on the followingdescription.

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented. Data processingenvironment 100 is a network of computers in which the illustrativeembodiments may be implemented. Data processing environment 100 includesnetwork 102. Network 102 is the medium used to provide communicationslinks between various devices and computers connected together withindata processing environment 100. Network 102 may include connections,such as wire, wireless communication links, or fiber optic cables.

Clients or servers are only example roles of certain data processingsystems connected to network 102 and are not intended to exclude otherconfigurations or roles for these data processing systems. Server 104and server 106 couple to network 102 along with storage unit 108.Software applications may execute on any computer in data processingenvironment 100. Clients 110, 112, and 114 are also coupled to network102. A data processing system, such as server 104 or 106, or client 110,112, or 114 may contain data and may have software applications orsoftware tools executing thereon.

Only as an example, and without implying any limitation to sucharchitecture, FIG. 1 depicts certain components that are usable in anexample implementation of an embodiment. For example, servers 104 and106, and clients 110, 112, 114, are depicted as servers and clients onlyas examples and not to imply a limitation to a client-serverarchitecture. As another example, an embodiment can be distributedacross several data processing systems and a data network as shown,whereas another embodiment can be implemented on a single dataprocessing system within the scope of the illustrative embodiments. Dataprocessing systems 104, 106, 110, 112, and 114 also represent examplenodes in a cluster, partitions, and other configurations suitable forimplementing an embodiment.

Device 132 is an example of a device described herein. For example,device 132 can take the form of a smartphone, a tablet computer, alaptop computer, client 110 in a stationary or a portable form, awearable computing device, or any other suitable device. Any softwareapplication described as executing in another data processing system inFIG. 1 can be configured to execute in device 132 in a similar manner.Any data or information stored or produced in another data processingsystem in FIG. 1 can be configured to be stored or produced in device132 in a similar manner.

Application 105 implements an embodiment described herein. Application105 implements a remotely usable function (remote) of an embodimentdescribed herein. Application 111 implements a locally usable function(local) or a natively usable function (native) of an embodimentdescribed herein. Application 134 implements a natively usable function(native) of an embodiment described herein. Applications 105 and 111 canbe used in a combination, applications 105 and 134 can be used inanother combination, and applications 105, 111, and 134 can be used inanother combination, to distribute certain functions of an embodiment.Application 105 implements an embodiment described herein. Input data109 is an example dataset that can be used with an operation ofapplication 105 in a manner described herein. Application 105 uses inputdata 109 to train a neural network, which can then be operated to makepredictions consistent with input data 109, in a manner describedherein.

Servers 104 and 106, storage unit 108, and clients 110, 112, and 114,and device 132 may couple to network 102 using wired connections,wireless communication protocols, or other suitable data connectivity.Clients 110, 112, and 114 may be, for example, personal computers ornetwork computers.

In the depicted example, server 104 may provide data, such as bootfiles, operating system images, and applications to clients 110, 112,and 114. Clients 110, 112, and 114 may be clients to server 104 in thisexample. Clients 110, 112, 114, or some combination thereof, may includetheir own data, boot files, operating system images, and applications.Data processing environment 100 may include additional servers, clients,and other devices that are not shown.

In the depicted example, data processing environment 100 may be theInternet. Network 102 may represent a collection of networks andgateways that use the Transmission Control Protocol/Internet Protocol(TCP/IP) and other protocols to communicate with one another. At theheart of the Internet is a backbone of data communication links betweenmajor nodes or host computers, including thousands of commercial,governmental, educational, and other computer systems that route dataand messages. Of course, data processing environment 100 also may beimplemented as a number of different types of networks, such as forexample, an intranet, a local area network (LAN), or a wide area network(WAN). FIG. 1 is intended as an example, and not as an architecturallimitation for the different illustrative embodiments.

Among other uses, data processing environment 100 may be used forimplementing a client-server environment in which the illustrativeembodiments may be implemented. A client-server environment enablessoftware applications and data to be distributed across a network suchthat an application functions by using the interactivity between aclient data processing system and a server data processing system. Dataprocessing environment 100 may also employ a service orientedarchitecture where interoperable software components distributed acrossa network may be packaged together as coherent business applications.Data processing environment 100 may also take the form of a cloud, andemploy a cloud computing model of service delivery for enablingconvenient, on-demand network access to a shared pool of configurablecomputing resources (e.g. networks, network bandwidth, servers,processing, memory, storage, applications, virtual machines, andservices) that can be rapidly provisioned and released with minimalmanagement effort or interaction with a provider of the service.

With reference to FIG. 2, this figure depicts a block diagram of a dataprocessing system in which illustrative embodiments may be implemented.Data processing system 200 is an example of a computer, such as servers104 and 106, or clients 110, 112, and 114 in FIG. 1, or another type ofdevice in which computer usable program code or instructionsimplementing the processes may be located for the illustrativeembodiments.

Data processing system 200 is also representative of a data processingsystem or a configuration therein, such as data processing system 132 inFIG. 1 in which computer usable program code or instructionsimplementing the processes of the illustrative embodiments may belocated. Data processing system 200 is described as a computer only asan example, without being limited thereto. Implementations in the formof other devices, such as device 132 in FIG. 1, may modify dataprocessing system 200, such as by adding a touch interface, and eveneliminate certain depicted components from data processing system 200without departing from the general description of the operations andfunctions of data processing system 200 described herein.

In the depicted example, data processing system 200 employs a hubarchitecture including North Bridge and memory controller hub (NB/MCH)202 and South Bridge and input/output (I/O) controller hub (SB/ICH) 204.Processing unit 206, main memory 208, and graphics processor 210 arecoupled to North Bridge and memory controller hub (NB/MCH) 202.Processing unit 206 may contain one or more processors and may beimplemented using one or more heterogeneous processor systems.Processing unit 206 may be a multi-core processor. Graphics processor210 may be coupled to NB/MCH 202 through an accelerated graphics port(AGP) in certain implementations.

In the depicted example, local area network (LAN) adapter 212 is coupledto South Bridge and I/O controller hub (SB/ICH) 204. Audio adapter 216,keyboard and mouse adapter 220, modem 222, read only memory (ROM) 224,universal serial bus (USB) and other ports 232, and PCI/PCIe devices 234are coupled to South Bridge and I/O controller hub 204 through bus 238.Hard disk drive (HDD) or solid-state drive (SSD) 226 and CD-ROM 230 arecoupled to South Bridge and I/O controller hub 204 through bus 240.PCI/PCIe devices 234 may include, for example, Ethernet adapters, add-incards, and PC cards for notebook computers. PCI uses a card buscontroller, while PCIe does not. ROM 224 may be, for example, a flashbinary input/output system (BIOS). Hard disk drive 226 and CD-ROM 230may use, for example, an integrated drive electronics (IDE), serialadvanced technology attachment (SATA) interface, or variants such asexternal-SATA (eSATA) and micro-SATA (mSATA). A super I/O (SIO) device236 may be coupled to South Bridge and I/O controller hub (SB/ICH) 204through bus 238.

Memories, such as main memory 208, ROM 224, or flash memory (not shown),are some examples of computer usable storage devices. Hard disk drive orsolid state drive 226, CD-ROM 230, and other similarly usable devicesare some examples of computer usable storage devices including acomputer usable storage medium.

An operating system runs on processing unit 206. The operating systemcoordinates and provides control of various components within dataprocessing system 200 in FIG. 2. The operating system may be acommercially available operating system for any type of computingplatform, including but not limited to server systems, personalcomputers, and mobile devices. An object oriented or other type ofprogramming system may operate in conjunction with the operating systemand provide calls to the operating system from programs or applicationsexecuting on data processing system 200.

Instructions for the operating system, the object-oriented programmingsystem, and applications or programs, such as application 105 in FIG. 1,are located on storage devices, such as in the form of code 226A on harddisk drive 226, and may be loaded into at least one of one or morememories, such as main memory 208, for execution by processing unit 206.The processes of the illustrative embodiments may be performed byprocessing unit 206 using computer implemented instructions, which maybe located in a memory, such as, for example, main memory 208, read onlymemory 224, or in one or more peripheral devices.

Furthermore, in one case, code 226A may be downloaded over network 201Afrom remote system 201B, where similar code 201C is stored on a storagedevice 201D. in another case, code 226A may be downloaded over network201A to remote system 201B, where downloaded code 201C is stored on astorage device 201D.

The hardware in FIGS. 1-2 may vary depending on the implementation.Other internal hardware or peripheral devices, such as flash memory,equivalent non-volatile memory, or optical disk drives and the like, maybe used in addition to or in place of the hardware depicted in FIGS.1-2. In addition, the processes of the illustrative embodiments may beapplied to a multiprocessor data processing system.

In some illustrative examples, data processing system 200 may be apersonal digital assistant (PDA), which is generally configured withflash memory to provide non-volatile memory for storing operating systemfiles and/or user-generated data. A bus system may comprise one or morebuses, such as a system bus, an I/O bus, and a PCI bus. Of course, thebus system may be implemented using any type of communications fabric orarchitecture that provides for a transfer of data between differentcomponents or devices attached to the fabric or architecture.

A communications unit may include one or more devices used to transmitand receive data, such as a modem or a network adapter. A memory may be,for example, main memory 208 or a cache, such as the cache found inNorth Bridge and memory controller hub 202. A processing unit mayinclude one or more processors or CPUs.

The depicted examples in FIGS. 1-2 and above-described examples are notmeant to imply architectural limitations. For example, data processingsystem 200 also may be a tablet computer, laptop computer, or telephonedevice in addition to taking the form of a mobile or wearable device.

Where a computer or data processing system is described as a virtualmachine, a virtual device, or a virtual component, the virtual machine,virtual device, or the virtual component operates in the manner of dataprocessing system 200 using virtualized manifestation of some or allcomponents depicted in data processing system 200. For example, in avirtual machine, virtual device, or virtual component, processing unit206 is manifested as a virtualized instance of all or some number ofhardware processing units 206 available in a host data processingsystem, main memory 208 is manifested as a virtualized instance of allor some portion of main memory 208 that may be available in the hostdata processing system, and disk 226 is manifested as a virtualizedinstance of all or some portion of disk 226 that may be available in thehost data processing system. The host data processing system in suchcases is represented by data processing system 200.

With reference to FIG. 3, this figure depicts a block diagram of anexample problem that can be solved with an illustrative embodiment.Matrix 302 is an example of input data 109 in FIG. 1.

Suppose that matrix 302 represents the example traffic dataset describedherein. As shown, matrix 302 comprises individual features for the 7days of a week and the 24 hours in a day. As can be seen, matrix 302 isquite sparsely populated. The problem with manual feature crossing isillustrated by matrices 304 and 306. When all hours of a day are crossedwith all days of a week, we do get some meaningful crosses but now alarger 168 column-wide matrix results that is even more sparse thanmatrix 302. Conversely, if we cross too many or unrelated features, asin matrix 306, we can achieve a smaller matrix but its cells are notrepresentative of matrix 302 to a desirable degree.

With reference to FIG. 4, this figure depicts a block diagram of anexample embedding in accordance with an illustrative embodiment.Configuration 400 depicts neural network 402 which comprises a number oflayers, such as layer 402A, layer 402B . . . layer 402X. feature crosses404 comprising feature crosses FC1, FC2 . . . FCn are embedded in layer402B as an example. Any number of layer embeddings can be formed usingfeature crosses in this manner. For example, feature crosses 406 aresimilarly embedded in layer 402X in a manner described herein.

With reference to FIG. 5, this figure depicts a block diagram of anexample application for feature engineering in neural networksoptimization in accordance with an illustrative embodiment. Application502 can be implemented as application 105 in FIG. 1.

Application 502 receives input data 504, which may be in the form ofinput data 109 in FIG. 1 or matrix 302 in FIG. 3. Component 506 forms anontology with forward materialization in a manner described herein.Component 508 constructs the vector space corresponding to the ontologyand performs a similarity search in a manner described herein. Component510 transitively reduces the graph using the transitive closure datastructures formed from the ontology graph.

Component 512 identifies clusters of discriminative nodes (features)remaining in the reduced graph. The clusters of discriminative nodesextracted from, or isolated in, the reduced graph form a set ofcandidate feature crosses. A candidate feature cross is a discriminativefeature cross that can represent at least a portion of the originaldataset with a desired degree of accuracy.

Component 514 computes an upper bound and a lower bound for a candidatefeature cross as described herein. Given the embedding range defined bythe upper and lower bounds, component 516 determines adimensionality—layer size of the embedding—for the specific featurecross. In one embodiment, component 516 reduces, or otherwise changes oralters, the dimensionality of the embedding to set a layer size.

In one embodiment, application 502 outputs set 518 of the candidatefeature crosses. In another embodiment, application 502 outputs set 518of the candidate feature crosses together with upper and lower bonds 520for each candidate feature cross. In another embodiment, application 502outputs set 518 of the candidate feature crosses together with (i) upperand lower bonds 520 for each candidate feature cross, (ii) a computedlayer size 522, or (iii) both (i) and (ii).

Outputs 518, 520, 522 or some combination thereof are usable toconfigure a neural network. For example, in one embodiment, application502 causes a neural network to embed a subset of candidate featurecrosses 518 according to layer embedding size 522. According to oneembodiment, application 502 further causes the configured neural networkto form model implementation 524 in a prediction engine, such as inprediction engine 107. In one embodiment, the configuration, theimplementation, or both, are automatic without human intervention. Inanother embodiment, the configuration, the implementation, or somecombination thereof is performed after a humanreview/adjustment/selection of the one or more outputs.

Once the neural network is configured and implemented, in oneembodiment, application 502 receives as feedback 526 informationcontaining the set of implemented feature crosses, which may overlapcompletely or partially with a subset of candidate feature crosses 518.For example, a user might configure some feature crosses from thecandidate feature crosses 518 and some feature crosses from the user'sown experience that are not in set 518 of candidate feature crosses.

Model implementation 524 produces model output 528, e.g., a prediction.In one embodiment, application 502 receives output 528 as feedback frommodel implementation 524 directly. In another embodiment, application502 receives output 528 after a user has applied adjustment 530 tooutput 528. Either way, application 502 uses feedback 526 and adjustedor unadjusted output 528 to retrain the neural network using a suitableknown machine learning technique.

A detailed description of a specific embodiment implementation forsolving a specific example problem are now provided. This description isnot intended to be limiting on all embodiments described herein. Thoseof ordinary skill in the art will be able to adapt the details from thisembodiment into other embodiments and such adaptations are contemplatedwithin the scope of the illustrative embodiments.

Given a semantic graph (e.g. an Ontology) we want to query whether eg:Ais related through eg:p to eg:D and list the derivation route. A ruleset to compute the transitive closure over a given relation can beexpressed in an API-agnostic fashion:

Statement is[urn:x−hp:eg/A,urn:x−hp:eg/p,urn:x−hp:eg/D]

Rule rule1 concluded(eg:A eg:p eg:D)<−Fact(eg:A eg:p eg:B)

As a reminder, transitive closure is the method by which, given adirected graph, the system determines if a vertex v is reachable fromanother vertex u for all vertex pairs (u, v) in the given graph.Reachable means that there is a path from vertex u to v. Thereachability matrix is called transitive closure of a graph.

This reachability matrix is a method for computing the cost ofinference. A transitive inference of the first-order denotes thematerialization of a→c via the explicit notion of a→b→c as having a costof a single node to traverse. The cost is unchanged if the traversednodes exist within the same sub-class. If the traversed nodes occurwithin the context of a grand-parent sub-class, the cost increases by afunction of the cosine similarity between the classes.

   def  transitiveClosure(V, graph):reach = [i[:]  for  i  in  graph] for  k  in  range(V):   for  i  in  range(V):   for  j  in  range(V):   reach[i][j] = reach[i][j]  or  (reach[i][k]  and  reach[k][j])  return  reach $\begin{matrix}{{graph} = \left\lbrack {\left\lbrack {1,1,0,1} \right\rbrack,} \right.} \\{{\left\lbrack {0,1,1,0} \right\rbrack,}} \\{{\left\lbrack {0,0,1,1} \right\rbrack,}} \\\left. \left\lbrack {0,0,0,1} \right\rbrack \right\rbrack\end{matrix}$ transitiveClosure(graph)

FIG. 6 shows a materialization of inference in accordance with anillustrative embodiment. The materialization of inference is shown inthis figure via transitive closure graph 604 (on the right side) againsta deep semantic graph 602 (on the left).

In the example, graph 602 demonstrates a semantic graph subset. Thehash-patterned nodes are entities in the graph connected via anrdfs:subClassOf or equivalent relationship to the clear-patterned nodes(parents). Graph 604 on the right shows the materialization ofinferences. Inference materialization is a form of transitive closure.The darker edges represent a transitive reduction of the semantic graph,meaning that a path is now formed directly from node A to node F (forexample). The ability to compute the semantic cost of each transitivereduction materialized within graph 604 is a feature made possible by anillustrative embodiment.

The function takes each materialized entity (λ) across the entire graphmultiplied the transitive closure value (x) multiplied by the cosinesimilarity between parent nodes (cos θ). A non-limiting example cosinesimilarity algorithm is shown for reference as follows—

${f(x)} = \left\{ {{\sum\limits_{i = 0}^{n}\; {\cos \mspace{14mu} \theta \mspace{14mu} \lambda_{i}x}},{{\&\mspace{11mu} x} < {0\mspace{14mu} x}},{{{\&\mspace{11mu} x} \geq {0{Cos}\; \theta}} = {\frac{\overset{\rightarrow}{a} \cdot \overset{\rightarrow}{b}}{{\overset{\rightarrow}{a}}\mspace{14mu} {\overset{\rightarrow}{b}}} = \frac{\Sigma_{1}^{n}a_{i}b_{i}}{\sqrt{\Sigma_{1}^{n}a_{i}^{2}}\sqrt{\Sigma_{1}^{n}b_{i}^{2}}}}}} \right.$

The output of each function exists within an approximately normaldistribution. The z-score is computed by taking the point-value of eachentity and subtracting the mean dividing the result by the standarddeviation of all values. The decision of how to treat z-scores issomewhat policy-based, but in general, eliminates any statisticallysignificant outlier. The skew of the distribution drives policy in thisrespect.

FIG. 7 depicts an output graph resulting from graph 604 in FIG. 6according to an embodiment. Graph 702 represents the output of theformula on the hypothetical sub-graph 604 in FIG. 6. Entities B, G(shown as line patterned) are concatenated into a single cross-feature.Without the benefit of the above provided formula, existingconcatenation strategies would likely include entity E to the detrimentof the trained model. The method likewise concatenates entities C, E, F(shown as hash patterned) into a single cross feature. Entity D is leftout as the cost computation is too high.

Moving on from a hypothetical example to a real-world graph, supposethat a Tensorboard visualization depicts placement of an ontology into avector space. FIG. 8A depicts an placement of an ontology into a vectorspace. FIG. 8B depicts the materialization of inferences for candidatefeature crosses. An example candidate feature cross is depicted by theline-patterned nodes in graph 852. The line-patterned nodes denote acluster that has been formed and represent candidate cross-features fora statistical model. The lower-bound of embedding is computed by summingthe instance data of all the entities that contribute to each cluster(cross feature) as noted herein.

With reference to FIG. 9, this figure depicts a flowchart of an exampleprocess for feature engineering in neural networks optimization inaccordance with an illustrative embodiment. Process 900 can beimplemented in application 502 in FIG. 5.

The application receives an input dataset (block 902). The applicationconstructs an ontology graph from the dataset (block 904). Theapplication performs forward materialization with inferences on thegraph (block 906)

The application constructs a vector space of entities (features) in theforward materialized graph (block 908). The application identifies oneor more concepts in the vector space (block 910). The applicationdetermines similarities between concepts (block 912).

The application constructs transitive closure data structures for a pairof entities linked by similarities in their concepts (block 914). Usinga transitive closure data structure, the application performs atransitive reduction (block 916). The application performs thetransitive reduction for one or more transitive closure data structures.

The application identifies clusters of remaining nodes/entities/featuresin the reduced graph resulting from the transitive reduction operations(block 918). A cluster is a candidate feature cross. The applicationcomputes an upper bound for the candidate feature cross (block 920). Theapplication computes a lower bound for the candidate feature cross(block 922). The application computes an embedding layer size for usingthe upper and lower bounds for the feature cross (block 924).

The application configures a layer in a neural network according to thecomputed embedding layer size (block 926). Any number of features may beembedded in any number of layers in this manner. The application trainsthe neural network whose layers are configured in this manner (block928). The application deploys the trained neural network in a predictionengine (block 930). The application may end process 900 thereafter.

In one implementation, the application further receives feedback fromthe deployment in the prediction engine (block 932). The applicationadjusts a feature cross, an embedding layer size, or both, according tothe feedback as described herein (block 934). The application endsprocess 900 thereafter.

Thus, a computer implemented method, system or apparatus, and computerprogram product are provided in the illustrative embodiments for featureengineering in neural networks optimization and other related features,functions, or operations. Where an embodiment or a portion thereof isdescribed with respect to a type of device, the computer implementedmethod, system or apparatus, the computer program product, or a portionthereof, are adapted or configured for use with a suitable andcomparable manifestation of that type of device.

Where an embodiment is described as implemented in an application, thedelivery of the application in a Software as a Service (SaaS) model iscontemplated within the scope of the illustrative embodiments. In a SaaSmodel, the capability of the application implementing an embodiment isprovided to a user by executing the application in a cloudinfrastructure. The user can access the application using a variety ofclient devices through a thin client interface such as a web browser(e.g., web-based e-mail), or other light-weight client-applications. Theuser does not manage or control the underlying cloud infrastructureincluding the network, servers, operating systems, or the storage of thecloud infrastructure. In some cases, the user may not even manage orcontrol the capabilities of the SaaS application. In some other cases,the SaaS implementation of the application may permit a possibleexception of limited user-specific application configuration settings.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, including but not limited tocomputer-readable storage devices as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

Embodiments of the present invention may also be delivered as part of aservice engagement with a client corporation, nonprofit organization,government entity, internal organizational structure, or the like.Aspects of these embodiments may include configuring a computer systemto perform, and deploying software, hardware, and web services thatimplement, some or all of the methods described herein. Aspects of theseembodiments may also include analyzing the client's operations, creatingrecommendations responsive to the analysis, building systems thatimplement portions of the recommendations, integrating the systems intoexisting processes and infrastructure, metering use of the systems,allocating expenses to users of the systems, and billing for use of thesystems. Although the above embodiments of present invention each havebeen described by stating their individual advantages, respectively,present invention is not limited to a particular combination thereof. Tothe contrary, such embodiments may also be combined in any way andnumber according to the intended deployment of present invention withoutlosing their beneficial effects.

What is claimed is:
 1. A method comprising: constructing, using aprocessor and a memory, a transitive closure data structure for a pairof features represented in a vector space corresponding to an inputdataset, wherein the transitive closure data structure comprises a setof entries corresponding to a set of paths in a graph of the vectorspace, the set of paths comprising all possible paths in the graphbetween a first feature in the pair and a second feature in the pair;reducing the transitive closure data structure by removing from thetransitive closure data structure a subset of the set of entries suchthat only a single entry corresponding to a single path remains in thetransitive closure data structure; forming a feature cross from acluster of features remaining in a reduced ontology graph resulting fromthe reducing the transitive closure data structure; and configuring alayer in a neural network to represent the feature cross, wherein theconfigured layer in the neural network causes the neural network toproduce a prediction that is within a defined accuracy relative to thedataset.
 2. The method of claim 1, further comprising: computing, usingthe processor and the memory, for the layer, an upper bound of layerembedding corresponding to the feature cross; and computing, for thelayer, a lower bound of the layer embedding corresponding to the featurecross, wherein a layer size of the layer for layer embedding is based onthe upper bound and the lower bound.
 3. The method of claim 2, whereinthe layer size is a computed average of the upper bound and the lowerbound.
 4. The method of claim 1, further comprising: transforming thedataset into a sparse matrix; constructing an ontology graphcorresponding to the sparse matrix, wherein the ontology graph comprisesa set of vertices corresponding to a set of features in the dataset; andperforming a forward materialization on a graph, wherein the forwardmaterialization adds an inference vertex to the set of vertices, andwherein the set of vertices including the inference vertex is used inconstructing the transitive closure data structure.
 5. The method ofclaim 1, wherein the single remaining path satisfies an efficiencyrequirement.
 6. The method of claim 5, wherein the efficiencyrequirement comprises a shortest distance between the first feature andthe second feature.
 7. The method of claim 5, wherein the efficiencyrequirement comprises a highest usage between the first feature and thesecond feature.
 8. The method of claim 1, wherein the reducing comprisesa transitive reduction algorithm.
 9. The method of claim 1, wherein thereducing the transitive closure data structure causes a subset ofvertices to be removed from the ontology graph, forming the reducedontology graph.
 10. The method of claim 1, further comprising:identifying a plurality of clusters in the reduced ontology graph, theplurality of clusters including the cluster, and wherein the clustercomprises at least two features such that a combination of the twofeatures is usable to distinguish between at least a first portion ofthe dataset and a second portion of the dataset.
 11. A computer usableprogram product comprising a computer-readable storage device, andprogram instructions stored on the storage device, the stored programinstructions comprising: program instructions to construct, using aprocessor and a memory, a transitive closure data structure for a pairof features represented in a vector space corresponding to an inputdataset, wherein the transitive closure data structure comprises a setof entries corresponding to a set of paths in a graph of the vectorspace, the set of paths comprising all possible paths in the graphbetween a first feature in the pair and a second feature in the pair;program instructions to reduce the transitive closure data structure byremoving from the transitive closure data structure a subset of the setof entries such that only a single entry corresponding to a single pathremains in the transitive closure data structure; program instructionsto form a feature cross from a cluster of features remaining in areduced ontology graph resulting from the reducing the transitiveclosure data structure; and program instructions to configure a layer ina neural network to represent the feature cross, wherein the configuredlayer in the neural network causes the neural network to produce aprediction that is within a defined accuracy relative to the dataset.12. The computer usable program product of claim 11, further comprising:program instructions to compute, using the processor and the memory, forthe layer, an upper bound of layer embedding corresponding to thefeature cross; and program instructions to compute, for the layer, alower bound of the layer embedding corresponding to the feature cross,wherein a layer size of the layer for layer embedding is based on theupper bound and the lower bound.
 13. The computer usable program productof claim 12, wherein the layer size is a computed average of the upperbound and the lower bound.
 14. The computer usable program product ofclaim 11, further comprising: program instructions to transform thedataset into a sparse matrix; program instructions to construct anontology graph corresponding to the sparse matrix, wherein the ontologygraph comprises a set of vertices corresponding to a set of features inthe dataset; and program instructions to perform a forwardmaterialization on a graph, wherein the forward materialization adds aninference vertex to the set of vertices, and wherein the set of verticesincluding the inference vertex is used in constructing the transitiveclosure data structure.
 15. The computer usable program product of claim11, wherein the single remaining path satisfies an efficiencyrequirement.
 16. The computer usable program product of claim 15,wherein the efficiency requirement comprises a shortest distance betweenthe first feature and the second feature.
 17. The computer usableprogram product of claim 15, wherein the efficiency requirementcomprises a highest usage between the first feature and the secondfeature.
 18. The computer usable program product of claim 11, whereinthe stored program instructions are stored in a computer readablestorage device in a data processing system, and wherein the storedprogram instructions are transferred over a network from a remote dataprocessing system.
 19. The computer usable program product of claim 11,wherein the stored program instructions are stored in a computerreadable storage device in a server data processing system, and whereinthe stored program instructions are downloaded over a network to aremote data processing system for use in a computer readable storagedevice associated with the remote data processing system, furthercomprising: program instructions to meter use of the computer usablecode associated with the request; and program instructions to generatean invoice based on the metered use.
 20. A computer system comprising aprocessor, a computer-readable memory, and a computer-readable storagedevice, and program instructions stored on the storage device forexecution by the processor via the memory, the stored programinstructions comprising: program instructions to construct, using aprocessor and a memory, a transitive closure data structure for a pairof features represented in a vector space corresponding to an inputdataset, wherein the transitive closure data structure comprises a setof entries corresponding to a set of paths in a graph of the vectorspace, the set of paths comprising all possible paths in the graphbetween a first feature in the pair and a second feature in the pair;program instructions to reduce the transitive closure data structure byremoving from the transitive closure data structure a subset of the setof entries such that only a single entry corresponding to a single pathremains in the transitive closure data structure; program instructionsto form a feature cross from a cluster of features remaining in areduced ontology graph resulting from the reducing the transitiveclosure data structure; and program instructions to configure a layer ina neural network to represent the feature cross, wherein the configuredlayer in the neural network causes the neural network to produce aprediction that is within a defined accuracy relative to the dataset.